Maximal covers of finite groups

Let $λ(G)$ be the maximum number of subgroups in an irredundant covering of the finite group $G$. We prove that if $G$ is a group with $λ(G) \leqslant 6$, then $G$ is supersolvable. We also describe the structure of the groups $G$ with $λ(G)=6$. Moreover, we show that if $G$ is a group with $λ(G) \leqslant 30$, then $G$ is solvable.

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